**Introduction**

Geometrical figures are the most used and important concept of Geometry in Mathematics. Geometry consists of a vast number of geometrical figures. For example- Triangle, Circle, Parallelogram, Cylinder, Rhombus, Polygon, Hexagon, Square, and many more.

Among other geometrical figures, Rhombus and Parallelogram are often misunderstood as the same, mainly by students. Rhombus is completely different from Parallelogram. The rhombus is a 2-D-shaped quadrilateral that consists of four vertices and four edges with equal sides. Rhombus has an area, perimeter, types, properties, and symmetries. It is also referred to as Equilateral Parallelogram.

On the other hand, as the name suggests, a Parallelogram is a quadrilateral of four vertices and four edges with opposite sides parallel and equal. It consists of area, perimeter, properties, and type. It is a special kind of two-dimensional polygon.

To understand the concept of these two figures, it is necessary to have a look at the basic concepts and differences between the two.

**Rhombus vs. Parallelogram**

Geometry consists of different kinds of quadrilaterals. A quadrilateral is a polygon with four sides. Like squares, rectangles, and other quadrilaterals, Rhombus and Parallelogram are also two of the well-defined quadrilaterals in geometry.

A rhombus is a quadrilateral with four sides of equal measure and equal length. The diagonals of the rhombus intersect at 90 degrees. It is also referred to as Equilateral Quadrilateral, Slanting Square, Diamond, Lozenge, and many other names. The sides of a rhombus are parallel to each other. Its diagonals are perpendicular to each other at the point of intersection. The rhombus is shaped like a diamond. Thus, it is in a diamond shape.

Whereas, parallelogram is also a four-sided quadrilateral, but the sides of a parallelogram are parallel to each other. Its diagonals bisect each other and make two congruent triangles. Congruent triangles are the triangles in which three of the sides and angles are corresponding to each other. In a parallelogram, the opposite angles are also equal in measure. The sum of the interior angles of the parallelogram is equal to 360 degrees. The set of parallel lines is connected at the endpoints and is equal. This type of quadrilateral is flat in shape.

In Euclidean Geometry, it is shown that every square, rhombus, and rectangle is a parallelogram. But not all parallelograms are square, rhombus and rectangle. That's why both rhombus and parallelogram differ from each other based on sides, area, perimeter, and other properties.

**Difference Between Rhombus and Parallelogram in Tabular Form**

Parameters of Comparison | Rhombus | Parallelogram |
---|---|---|

Definition | Rhombus is a diamond-shaped four-sided quadrilateral with four sides of equal measure. | A parallelogram is a flat-shaped four-sided quadrilateral with opposite sides parallel. |

Diagonals | The diagonals of the rhombus bisect each other at 90 degrees, forming an equilateral triangle. | The diagonals of the parallelogram bisect each other and form two congruent triangles. |

Length | The length of the four sides of the rhombus is the same. | Only opposite sides of the parallelogram are equal in length. |

Opposite Angles | In a rhombus, the sum of opposite angles is always 180 degrees. | In a parallelogram, the sum of opposite angles is 180 degrees. |

Area | Area of rhombus = (d1 x d2) / 2, where d1 and d2 are diagonals. | Area of parallelogram = a x h, where a is the base and h is the height. |

Perimeter | Perimeter of rhombus = 4 x a, where a = side. | Perimeter of parallelogram = 2(a + b), where a = side and b = base. |

**What is Rhombus?**

In Euclidean Geometry, a rhombus is a quadrilateral with four sides of equal measure and of equal length. Apart from the sides, the opposite angles of the rhombus are equal. It is stated that every rhombus is a parallelogram, but not every parallelogram is a rhombus. This is because every opposite side of a rhombus is parallel to each other. That is why every rhombus is considered a parallelogram.

Rhombus has a special kind of shape and is used in our daily lives. There are various rhombus-shaped objects. For example- Kaju Katli, Diamond, Jewellery, Kites, etc.

There are a few notable properties of the rhombus as follows:-

1. All sides of the rhombus are equal in length and measure. Here, AB=BC=CD=DA.

2. The diagonals of the rhombus bisect each other at 90 degrees. Here, CA and DB bisect each other at 90 degrees.

3. The opposite angles of a rhombus are congruent or equal to each other.

4. The sum of two adjacent angles is supplementary, i.e. 180 degrees. Supplementary angles are the angles whose sum is 180 degrees.

5. Circumscribing a circle is impossible around a rhombus.

6. Rhombus is not a cyclic quadrilateral.

7. The sum of the interior angles of the rhombus is 360 degrees.

There are two major formulas for Rhombus:-

1. **Area**- The area of the rhombus is the region surrounded by the equal sides. It can be found in two ways:-

a. When the base and height of the rhombus are given,

Area b x h sq. Units

b. When diagonals of the rhombus are given,

Area= [d1xd2] upon 2 sq. Units where d1 and d2 are diagonals. D1 is the length of diagonal 1, and d2 is the length of diagonal 2.

The area of the rhombus can be measured using trigonometric ratios. The formula is:-

Area of a rhombus= b2x sin [A] sq. Units

2. **Perimeter**- Square has 4 sides. Similarly, all 4 sides of the rhombus are equal. To find the perimeter of the rhombus, the formula is:-

P= 4a, where a is the side of the rhombus.

The rhombus also has two lines of symmetry. The line of symmetry is the line that divides the rhombus into two halves. The lines of symmetry in a rhombus originate from its diagonals. So, it can be said that the diagonals of a rhombus are its line of symmetries.

**EXAMPLES**

Find the area of a rhombus with diagonals of 16 m and 18 m.

**Solution:** Given:

Area of a rhombus (A) = (d1 * d2) / 2 A = (16 m * 18 m) / 2 A = (288 m²) / 2 A = 144 m²

Thus, the area of the rhombus is 144 m².

Diagonal 1 (d1) = 16 m

Diagonal 2 (d2) = 18 m

Find the area of a rhombus having a base of 12 m and a height of 16 m.

**Solution:** Given:

Area of a rhombus (A) = b * h A = 12 m * 16 m A = 192 m²

Thus, the area of the rhombus is 192 m².

Base (b) = 12 m

Height (h) = 16 m

Find the perimeter of a rhombus having a side length of 6 cm.

**Solution:** Given:

Perimeter of a rhombus (P) = 4 * a P = 4 * 6 cm P = 24 cm

Thus, the perimeter of the rhombus is 24 cm.

Side length (a) = 6 cm

Find the perimeter of a rhombus having a side length of 12 inches.

**Solution:** Given:

Perimeter of a rhombus (P) = 4 * a P = 4 * 12 inches P = 48 inches

Thus, the perimeter of the rhombus is 48 inches.

Side length (a) = 12 inches

**What is Parallelogram?**

A parallelogram is a special type of quadrilateral with parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measurement. As mentioned earlier, all rhombuses are parallelograms, but not every parallelogram is a rhombus. This is because opposite sides of a parallelogram are equal, but all sides of a rhombus are equal.

A parallelogram is flat in shape and is two-dimensional. There are many examples of items in the shape of parallelograms in real life. For example- the roof of a hut, notebooks, tiles, tennis courts, tables, cell phones, buildings, and many more.

Diagonals connect the opposite corners of the parallelogram. To find the area and perimeter of the parallelogram, there are two formulas:-

1. Area of parallelogram: The area of the parallelogram is the region surrounded by the equal sides. It can be found with the given formula:-

a= b x h, where b is base and h is height

If the height of the parallelogram is not given, then we can use the trigonometric formula:-

Area ab sin [x] where a and b are the lengths of adjacent sides of the parallelogram and x is the angle.

2. Perimeter of parallelogram= The sum of side and base is the perimeter of the parallelogram.

2 [a+b], where a= side and b= base.

**EXAMPLES**

**Find the area of the parallelogram with a base of 4 cm and a height of 5 cm.**

Solution:

Area of the parallelogram = B x H square units = 4 cm x 5 cm = 20 square cm

Thus, the area of the parallelogram is 20 cm².

Base (B) = 4 cm

Height (H) = 5 cm

**The angle between any two sides of a parallelogram is 90 degrees. If the lengths of the two adjacent sides are 3 cm and 4 cm, find the area.**

Solution: Let a = 3 cm and b = 4 cm Angle (X) = 90 degrees

Area = a x b x sin(X) = 3 cm x 4 cm x sin(90 degrees) = 12 cm² x 1 = 12 square cm

Thus, the area of the parallelogram is 12 cm².

**Find the perimeter of a parallelogram with a base and side lengths of 10 cm and 5 cm.**

Solution: Base (B) = 10 cm Side (S) = 5 cm

Perimeter (P) = 2(B + S) units = 2(10 cm + 5 cm) = 2(15 cm) = 30 cm

Thus, the perimeter of the parallelogram is 30 cm.

Euclidean geometry includes parallelogram law. There are two parallelogram laws:-

1. Parallelogram Law of Addition- This law states that the sum of the squares of the length of all the sides of a parallelogram is equal to the sum of the squares of the length of the two diagonals.

2. Parallelogram Law of Vectors- This law states that when two vectors are acting simultaneously, both in magnitude and direction, by the adjacent sides of a parallelogram drawn from the same point, then the resultant vector is represented by the diagonal of the parallelogram and vice-versa.

**Some notable properties of the parallelogram are:-**

- The opposite sides of a parallelogram are parallel and equal in length.
- The opposite angles of a parallelogram are equal.
- The consecutive angles of a parallelogram are supplementary. Here, A + D = 180°.
- The opposite sides of a parallelogram are congruent. Here, AB is congruent to DC, and AD is congruent to BC.
- The opposite angles of a parallelogram are congruent. (It seems like you left this statement incomplete. You can add something like "Here, angle A is congruent to angle C, and angle B is congruent to angle D.")
- The diagonals of a parallelogram bisect each other.
- Each diagonal of a parallelogram separates it into two congruent triangles.

**Main Differences Between Rhombus and Parallelogram (In Points)**

- A rhombus is a quadrilateral with four sides of equal measure and length, whereas a parallelogram is a special type of polygon with parallel sides.
- The word Rhombus is derived from the Latin word 'rhombus', which means a spinning top. On the other hand, the word Parallelogram is derived from the Greek word 'parallelogrammon', which means bounded by lines.
- Rhombus has many different names, like equilateral quadrilateral, diamond, and slanting square. In contrast, a parallelogram can be named as a two-dimensional polygon.
- The rhombus has a higher degree of symmetry due to congruent sides and angles. On the other hand, a parallelogram has a lower degree of symmetry than it.
- Ancient Greek Mathematicians like Pythagoras and Euclid studied rhombus and its properties. In contrast, Parallelogram was studied by ancient Egyptians for construction and land purposes.
- The rhombuses can be used as an example in literature to describe a man's life as balanced. On the other hand, a parallelogram can be used to symbolize the life of two people who never meet i.e., parallel sides.
- In physics, a rhombus is studied in crystallography, which is an essential unit. But parallelogram is mainly studied in the mechanics subject in physics.
- The theory in Euclidean Geometry states that Rhombus can be considered both a parallelogram and a square in some special cases. But, a parallelogram can be only considered a square in some rare cases.

**Conclusion**

Rhombus and Parallelogram are considered quadrilaterals, but there is a major difference in the length of sides, angles, properties, diagonals, shapes, intersections, and bisections. Just like squares, rhombuses also have equal sides with equal measurements. In comparison, a parallelogram has only equal and opposite sides. Thus, we can conclude that a rhombus is a special type of parallelogram and that every rhombus can be a parallelogram but not vice-versa.